A proof of the Scott–Wiegold conjecture on free products of cyclic groups

Abstract

Problem 5.53 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 12th Edition, Russian Academy of Sciences, Novosibirsk, 1992) (contributed by Wiegold, attributed to Scott) asks whether a free product of three (finite) cyclic groups can be normally generated by a single element. We give a proof of the conjectured negative answer, and an application to Dehn surgery on knots: if Dehn surgery on a knot is S 3 gives a connected sum, then all but at most 2 of the connected summands are Z -homology spheres, and hence (by a result of Valdez and Sayari) the number of connected summands is at most 3.